What we have learnt about in the first Topic are large-scale
(macro) properties of the Earth. They are also mostly "static"
properties - that is, properties that pertain to descriptions of the Earth,
even if it were not moving in space, and even if it were internally dead.
In fact, the Earth experiences a variety of complex motions in space and
is internally very much alive. These are the subjects of our next Topics.

Two aspects of the Earth's motion in space
are well known. First, the Earth rotates on its axis once every 24 hours. In
fact, the rotation rate is not exactly 24 hours, and varies by a small amount
on a daily basis; a matter of a few milliseconds. Additionally, it is slowing
down due to tidal friction (see Tides later) so that the length of a day has
actually increased over the history of the Earth. Second, the Earth rotates
around the Sun once a year. In fact, that is what is meant by a year.

The Earth experiences three dominant periodic motions of
much longer periods that are not readily perceptible. Each of these give
insight into the interior structure of the Earth or has a direct effect
on the Earth's climate.

2.1.1. Axial Precession

The rotational axis of the Earth is tilted with respect
to the normal to the plane of the ecliptic, presently at an angle of about
23.5°. That is, it doesn't stand up straight but leans over a bit. Because
of the oblate shape of the Earth and the tilt, the Sun exerts a gravitational
pull that differs in force at the two poles. The effect is to attempt to
set the Earth up straight - i.e., to try to right the axis to be normal
to the ecliptic. This would be a least energy configuration. In fact, the
"righting force" has the effect of causing the rotational axis
to execute a circular, wobbling motion, like a spinning top.

To understand the motion imagine a rod through the Earth's
axis of rotation. Hold the rod at its middle point (the center of the Earth)
and make one end move around in a circle. That's what the precession describes.
The motion goes through a complete circle in a period of about 26,000 years.
Why 26,000 years? This period is governed by the magnitude of the gravitational
field and the tilt angle (23.5°), but also the "dynamic elipticity",
H.

The dynamic elipticity is the ratio of the Earth's "moments
of inertia". The concept of moment of inertia is related to the more
familiar concept of inertia.

Inertia - tendency of a body to stay at rest, or retain
its current motion.

Moment of Inertia - tendency
of a force to cause a body to rotate, or the tendency of a body to remain
rotating once rotating.

Moment of Inertia must always be defined about an axis.

The Earth can be considered to have two moments of inertia
- one defined about the polar axis (C); the other about an axis drawn through
the equatorial plane (A).

The ratio is called the
dynamic elipticity, and controls the rate of motion of the rotational axis
in space. Remember that the ratio of axial lengths describing the Earth's
shape is its elipticity and could be thought of as its static elipticity.

Considering the problem in an inverse way, if we knew the
rate of motion we should be able to calculate the dynamic elipticity, H.
The rate of motion is known from the "precession of the equinoxes"
- the fact that over history different constellations have been visible
overhead at different times in the past. At the Spring Equinox, Pices is
directly overhead at the equator, while it has been recorded that in 120
B.C. Aires was overhead at the Spring Equinox. The 26,000 year period can
be estimated that way although more accurate present day calculations give
a better figure.

Using 26,000 years we can deduce that

and that

where M is the mass of the Earth, and a is the equatorial
radius. What does this tell us? We can also calculate moments of inertia
for various bodies.

Becausefor the Earth, it is somewhere between a point and a
solid. Hence it is likely that its density increases toward the center.

If we suppose that the Earth has two basic components -
half being a dense core and the other half a less dense outer shell - we
can also calculate moments for these bodies.

Moment of Inertia

2

0.367

3

0.339

<------- Closer to observed

4

0.338

If , a figure appropriate for the outer shell, then -
a fairly good estimate of average density of the Earth's core.

So, as in the topics we dealt with learning about the Earth's
macroscopic (but static) properties we have

Observation

Theory

Deduced Property

26,000 year precession

Rotational dynamics

Internal density distribution

2.1.2 Changes in Tilt

While the inclination of the Earth's rotational axis is
presently 23.5° it has moved a little in the past. Like the precession
of the equinoxes that reflects the circular motion of the rotational axis,
the tilt changes periodically also - from about 24.5° to 21.5° over
a period of about 40,000 years. So as the rotational axis wobbles, it also
tilts back and forth. This motion provides no insight into the Earth's internal
properties but it turns out to be very important as an influence on the
Earth's climatic changes.

2.1.3 Changes in the Earth's Orbit

The Earth revolves around the Sun at a full revolution
rate of 365 days. The shape of the orbit is eliptical with the Sun at one
focus. This orbital state too is not static. The elipticity of the orbit
changes, also in a periodic way, with this period being the longest - at
around 100,000 years. The elipticity changes from 0.01 to 0.007; i.e., by
6%, quite a large amount.

In summary, these three motions can be thought of as a
100,000 year stretch of the orbital elipse, a 40,000 year tilt
of the rotational axis, together with a 26,000 year wobble of
the rotational axis.Wobble and stretch are actually
coupled - if one changes the other is also forced to change along with it.

2.2 Climate changes and the earth's motion in space

These periodic changes give a significant component to
the fluctuations of the Earth's long-term climate. The change from glacial
to interglacial climates in the Earth's past largely follow these cycles.
Temperature in the past can be deduced from the fossil record, tree rings,
growth rings in corals and the chemistry of sediments deposited in the deep
oceans among other proxies.

The vertical graph below shows changes in past temperature inferred
from the abundances of fossils that represent cool water and warm water favoring
assemblages. The horizontal graph is temperature derived from oxygen isotopes
in sediments. Particularly in this one, the eye can pick out variations on 100,
40, and 20 thousand year periods.

The horizontally oriented chart in particular shows the
dominance of the 100 thousand year changes. By this we mean that the amplitude
of the inferred temperature swings are greater at 100 thousand year periods
than at other periods, You can see that the 20 thousand year variations
seem to ride on the 100 thousand year variations at and have only about
1/3 the amplitude at most. We say that there is more power
in the 100 thousand year periods. An analysis of proxy temperature records
from further back in the past shows that the dominace of the 100 thousand
year period has only been with us for that last million years. From 3 to
1 million years ago the 40 thousand year period was dominant and prior to
that the 20 thousad year period dominated. What's more the change from one
dominant period to the other was very abrupt. The changes from one dominant
period to another is one of the great misteries of climate science today,
especially the fact that these changes happen so rapidly.

Thelast chart shown above is one
of the most revealing record of past climate that has come available in
the last decade. It shows the history of temperatare variations and carbon
dioxide variations for the past 160,000 years as recorded in an ice core
taken from the summit of the Greenland ice dome. The record is very high
resolution because the ise records annual bands associated with changing
precipitation during the year much as trees have annual growth rings. Several
things can be quickly seen. One is that, for most of the last 160,000 years
the temperature has been quite a bit colder than it is today - the zero
on the temperature scale is today's temperature, the readiings show differences
from today's temperature. The last time the Earth's temperature was similar
to today's is more than 100,000 years ago. Our society seems to have come
about in a fairly unusual cliamtic setting. A second thing is that there
is a very large amoint of temperature fluctuation at high frequency during
the cold periods. These fluctuations if viewed in detail have periods of
a couple of thousand years. Clearly they are well short of the periods assocaited
with the tilt, wobble and stretch of planetary. Their origin is unknown.
A third aspect of this record is that there are very abrupt transitions
to very different climate states. At around 130,000 years and again at around
20,000 years the temperatures rocket up from cold to warm conditions. These
changes, again looked at in great detail, are actually found to take place
over a time period of less than a hundred years. In other words, the Earth's
climate appears to be capable of switching from cold to warm state in a
period comparable to that of a human life span. This is a startling and
unexpected new discovery. Such rapid changes cannot be caused by orbital
motion changes in any simple way that we know right now.

The other feature one sees on these curves is a high degree
of correlation between top curve showing carbon dioxide content and the
lower one showing temperature. Both are taken from the ice core data and
show that through natural variation (not human induced) temperature and
carbon dioxide levels go hand in hand. This is one of the key pieces of
evidence used to suggest that if hunam activity were to increase carbon
dioxide levels (as it has in post-industrial times) then global temperatures
are likely to increase also.

2.3 Another regular motion - Chandler Wobble

In addition to the three large scale motions we have discussed,
the Earth experiences a nodding (or nutation) called Chandler Wobble. This results
because the Earth is not rotating about its axis of figure, or axis of maximum
moment of inertia. The point on the Earth's surface where the rotational axis
intersects, revolves around the axis of figure in yet another periodic motion,
although, as can be seen from the attached map, it has a random component overprinting
a regular rotation.

The period of the nodding can be deduced and it should
be no surprise that it too is related to differences in the Earth's moments
of inertia C-A. If the Earth were a purely rigid solid the Chandler period
would be 305 days, but the observed period is over 400 (430).

The slower period is due to the fact that the Earth actually
yields a little to gravitational torques because it is not a purely rigid
solid.

Causes of the exciting and damping of Chandler wobble have
been debated for many yields. Damping may be due to motion in the fluid
core, but the reason the motion does not eventually evolve toward a rotational
axis coinciding with the axis of figure is not known. Something always excites
the motion by kicking the rotational axis away from the axis of symmetry.

Part of the answer to this question is that they are very
regular. When we say that the tilt, wobble and stretch of the orbital
motions have periods of 20, 40 and 100 thousand years that is usually taken
to mean that these are regular periods. And in fact that are regular, but
only approximately. There is a spread in period around 20 thousand (say)
and it is not possible to predict exactly what the period will be.
The variation is quite small so for most purposes in analyzing the past
record the periods can be taken to be very nearly regular.

This is not the case for the tilt variations of Mars, however.
Mars has a much greater range of tilt variations from around 11 degrees
to 38 degrees. And there is no well defined period. the motion of Mars'
tilt axis is an unpredictable wobbling around. It always lies between 11
degrees and 38 degrees but within that range it is very difficult to predict
what the titlt will be at any time. Most ofthe planets have somewhat chaotic
tilt variations. It is believed that the Earth may also have had a chaotic
tilt variability in the past but that the presence of the Moon has acted
to gravitationally stabilize the tilt by "holding it steady".

A quick look at the Chandler Wobble path leads to the conclusion
suggests a somewhat unpredictable behavior. Again, like most such systems
the variation has bounds -- the rotation pole and the axis of figure never
get beyond a certain distance apart and the period at which one circles
the other is always about 430 days, but never exactly predictable.

All of the motions then have some degree of unpredictability
associated with them. Mars' tilt is highly unpredicitable, Earth's precession
for instance is really quite predictable. In asking the question -- are
the motions chaotic -- the answer varies with the planet and the particular
motion involved.

2.5 Length of Day Variations.

While it might seem fairly incredible the length of the
day is changing all the time -- not by an amount that anyone would notice,
but by amoints that are easy to detect instrumentally. The changes are only
a few milliseconds (a millisecond is one thousandth of a second) but they
are very easy to measure. The record below shows that there are very regular
variations and also very irregular variations. All of the changes must in
some way be caused by the re-distribution of mass over the surface and with
in the Earth. Seasonal changes (curve d) are caused by the change in ice
volume and relative amounts of water in the atmosphere from summer to winter.
Longer period changes and shorter period changes (curves c and e respectively)
are more difficult to explain. Some are caused by the phase of the El Nino
-- the Earth rotates differently in an El Nino year than in a La Nina year
because these phenomena change the distribution of wet and dry parts of
the world. Even longer period changes must be caused by processes in the
deep Earth that move masses around such as subduction of large slabs of
lithosphere. The cause of many of the longer period signals is not well
understood.

2.6 Marine Tides

Every one who has been to the beach or spent any time near
a shore line knows something about the behavior of tides. They appear to
the beachgoer as the advance and retreat of the high water line. In fact,
we all know that what is happening is that the water level is rising and
falling throughout the day. Tides are a very predictable feature of the
Earth and result from the interactions of the Earth, Moon and Sun, together
with the rotation of the Earth and Moon.

The basic features of tides are well known:

In most places there are two high and two low tides each day.

The height of the high tide varies, having two maxima in a
month.

The figure below shows these features in tide gage readings (the height
of the tide) at several locations around the world for the same one month
period. You can see that in some places there are two high and two low tides
a day (called semi-diurnal) and in others there is only one (called diurnal).
Note also that the maximum height of the tide changes throughout the month.
Look carefully and you will also see that the time at which the tide reaches
a maximum advances a little each day.

Tides are very different from place to place throughout the
Earth.

The figure below shows how the tidal heights actually vary throughout
the world. The pattern is very complex in detail and cannot be explained by
the simple theory we cover in the following. Note that the animation at the
very end of this section shows how this complex pattern varies through a month.

These basic features can be explained by the "equilibrium"
theory of tides.

Tides are caused by the gravitational attraction of the
Moon and Sun. The Moon's effect is about 54% of the total - gravitational
force is proportional to the masses of the interacting bodies, but inversely
proportional to the square of the distance separating the bodies
so, although the Sun outweighs the Moon by a factor of 10 million the fact
that it is almost 400 times further away than the Moon gives the Moon the
gravitational edge.

Water is very weak and will respond to gravitational attraction
or "pull" by trying to move in the direction of the force. So
the water covering the Earth distorts by making a bulge toward the Moon.

As the Earth rotates, points on the surface move in and
out, under the bulge, and hence experience high and low tides. But there's
a problem with this model of tides; it can't account for there being two
high and two low tides each day. It can only explain one high and one low
as the Earth rotates beneath the one bulge. Our simple model is incomplete
because it doesn't take account of the rotation of the Earth/Moon system.
We generally think of the Moon rotating around the Earth on a monthly basis
while, in fact, the Earth and Moon rotate around each other. The gravitational
attraction acts to "connect" the two bodies and they rotate in
unison. If they were of equal mass they would rotate about their mid-point
like a baton thrown in the air by a majorette. Because the Earth's mass
is much greater than the Moon's the point about which Earth and Moon rotate
together (CM) is near the Earth's surface.

The animation that is available at http://plabpc.csustan.edu/general/tutorials/PlanetaryMotion/PlanetaryMotion.htm
shows you how the motion of two bodies rotating around each other is affected
by the relative sizes of the masses. You can use the sliders to change the masses.
You will see how making the masses more nearly equal causes the center of rotation
to be somewhere about half-way between the bodies, while if one mass is much
greater than the other the center of mass, and hence the rotation point is actually
within the more massive body. That's the way it is with the Earth/Moon system.

The critical effect this has is that it gives rise to a
second force - the centrifugal force - that acts in the opposite direction
and wants to create a water bulge on the opposite side of the Earth, away
from the Moon.

These two forces, in combination, give rise to equal size
bulges on both sides of the Earth, and explain why there are usually two
high and two low tides a day. This explains one of the most basic observations
about tides.

We also know that the basic pattern of daily tidal variations
changes from place to place. Usually there are two high and two low tides
but the height of the two "highs" is usually not the same. In
some places there is only one high tide in a day and in others the signal
appears mixed between the two. The gross features of this variable behavior
are explained by remembering that the Earth's rotational axis is tilted
with respect to the plane of the ecliptic by 23.5° as we know
from our previous studies, and the Moon is inclined at about 5°. The
net effect is that the tidal bulge is 28.5° off the equator.

That's almost everything. All that's left is to explain
why the height of the tides change during a month.

This is due to the combined effect of Moon and Sun. When
the two are aligned the gravitational attractions of the two reinforce and
create very high tides. This occurs when the Moon is "full" and
"new". When the Moon and Sun are maximally out of line, at the
Moon's "quarter" positions, the tides are lowest. The high tides
are called Spring tides and the lowest are rip tides. We often hear news
reports that describe the destructive effects of storms that surge at the
same time as a Spring tide - just about the worst combination.

In reality, tides are even more complex than have been
described above and we need a more complete theory (or model) to explain
their exact behavior. Missing from our simple model are the continents (our
equilibrium model has a fully water enshrouded Earth) and the topographic
effects of the seafloor including bays and inlets. Even the friction between
the seafloor and the water layers must be factored in. These effects are
very complex and no fully complete model of tides actually exists. All are
approximations.